10 REPRESENTATIONS OF ALGEBRAS OF CONTINUOUS FUNCTIONS 393
the Hilbert sum E,, © Fj$ 1 © - © Fj$k © - . It is clear from the construction
that
for each n\ since E is the Hilbert sum of the Ffc, it follows that E is also the
Hilbert sum of the Ek (6.4.2), and the subspaces Ek and measures ^k therefore
satisfy the required conditions.
Put jufc = gk fa, and let Sfc be the set of points t e K such that gk(t) > 0.
Evidently we may assume that the sequence (Sfc) is decreasing and that each
Sk is universally measurable (13.9.3). Put M! = K S2 and M* = Sk Sk+l
for k ^ 2; also put M^ = (] Sk. For 1 g / ^ fc, let H& = r(<pMk)(EŁ). Then
fcfci
it follows from (15.10.6) that the restrictions of T to the k subspaces
Hik (I <; iŁk) are equivalent, and we put Gfc = H1Jk © * © Hfck . Clearly Gk
is the Hilbert sum of the subspaces Hik (1 g f 5Ł fe). Similarly, put Hioo =
T(^M J(Ef) for each / ^ 1 . The restrictions ofTto all the subspaces Hioo (/ ^ 1)
are equivalent, and we denote by G^ the Hilbert sum of the Hfoo for all
i ^ 1. Then it is clear that E is the Hilbert sum of the Gk (k ^ 1) and G^ . It
can be shown (Problem 5) that the measures tik are determined up to equiv-
alence, and hence that the Sfc (and consequently the Mk) are determined up to
a /^-negligible set. The subspaces Gfc = T((pMj)(E) are uniquely determined
by T. The restriction of T to Gk (resp. to G^) is said to have multiplicity k
(resp. infinite multiplicity). A representation of multiplicity 1 is therefore
topologically cyclic.
(15.10.10) Since the measures fik are determined only up to equivalence
(1 5.1 0.7), we may suppose that fik = (p$k ju^ It follows that, up to equivalence,
the most general representation u\~+T(u) of ^C(K) on a separable Hilbert
space E may be described as follows :
Consider a positive measure v on K, and a decreasing sequence
(Sj)ig</«0 of universally measurable subsets of K, where co is either a
positive integer or + oo. The space E is the Hilbert sum of the Hilbert spaces'
Lc(K, cpSk - v); each of these spaces is stable under T9 and the restriction of
T(u) to Lc(K, (p$k v) is multiplication by u.
PROBLEMS
1. Let K be a compact space, E a Hilbert space, and (jc, y) \> /x-x, y a continuous sesquilinear
mapping of E x E into the Banach space MC(K) of complex measures on K. Show that,x.